A heat pump operates between temperature ranges of 100 C and 250 C. Find its COP. What will be its efficiency if the heat engine operates in the same temperature range?
A heat pump operates between temperature ranges of 100 C and 250 C. Find its COP. What will be its efficiency if the heat engine operates in the same temperature range?
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Question
![### Understanding Heat Pump and Heat Engine Operations
#### Problem Statement
1. A heat pump operates between temperature ranges of 100°C and 250°C. Find its COP (Coefficient of Performance).
2. What will be its efficiency if the heat engine operates in the same temperature range?
#### Detailed Explanation
**1. Coefficient of Performance (COP) of a Heat Pump:**
The COP of a heat pump, which is a measure of its efficiency, can be calculated using the temperatures at which it operates. The formula for the COP (Heating Mode) is:
\[
COP_{HP} = \frac{T_H}{T_H - T_C}
\]
where \(T_H\) is the high temperature (in Kelvin) and \(T_C\) is the low temperature (in Kelvin).
First, we need to convert the given temperatures from Celsius to Kelvin:
\[
T_H = 250°C + 273 = 523 \text{ K}
\]
\[
T_C = 100°C + 273 = 373 \text{ K}
\]
Now, using the formula:
\[
COP_{HP} = \frac{523}{523 - 373} = \frac{523}{150} \approx 3.49
\]
So, the COP of the heat pump is approximately 3.49.
**2. Efficiency of the Heat Engine:**
The efficiency \( \eta \) of a heat engine operating between the same temperature ranges is given by the Carnot efficiency formula:
\[
\eta = 1 - \frac{T_C}{T_H}
\]
Substituting the values:
\[
\eta = 1 - \frac{373}{523} \approx 1 - 0.713 \approx 0.287
\]
So, the efficiency of the heat engine is approximately 28.7%.
These calculations provide a foundational understanding of the efficiency metrics for both heat pumps and heat engines operating within specified temperature ranges. Such knowledge is crucial for designing and evaluating thermal systems in various engineering applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76dd9541-7b4f-4516-89bb-660714f6df81%2Fafa08f9e-d142-4ba1-a3a4-c1ed1982c6bc%2Fmpsmglc_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Heat Pump and Heat Engine Operations
#### Problem Statement
1. A heat pump operates between temperature ranges of 100°C and 250°C. Find its COP (Coefficient of Performance).
2. What will be its efficiency if the heat engine operates in the same temperature range?
#### Detailed Explanation
**1. Coefficient of Performance (COP) of a Heat Pump:**
The COP of a heat pump, which is a measure of its efficiency, can be calculated using the temperatures at which it operates. The formula for the COP (Heating Mode) is:
\[
COP_{HP} = \frac{T_H}{T_H - T_C}
\]
where \(T_H\) is the high temperature (in Kelvin) and \(T_C\) is the low temperature (in Kelvin).
First, we need to convert the given temperatures from Celsius to Kelvin:
\[
T_H = 250°C + 273 = 523 \text{ K}
\]
\[
T_C = 100°C + 273 = 373 \text{ K}
\]
Now, using the formula:
\[
COP_{HP} = \frac{523}{523 - 373} = \frac{523}{150} \approx 3.49
\]
So, the COP of the heat pump is approximately 3.49.
**2. Efficiency of the Heat Engine:**
The efficiency \( \eta \) of a heat engine operating between the same temperature ranges is given by the Carnot efficiency formula:
\[
\eta = 1 - \frac{T_C}{T_H}
\]
Substituting the values:
\[
\eta = 1 - \frac{373}{523} \approx 1 - 0.713 \approx 0.287
\]
So, the efficiency of the heat engine is approximately 28.7%.
These calculations provide a foundational understanding of the efficiency metrics for both heat pumps and heat engines operating within specified temperature ranges. Such knowledge is crucial for designing and evaluating thermal systems in various engineering applications.
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